If it were mine.... there would be no question in my mind about keeping both trunks... they are marvelous. The tree is well grown out, so there doesn't appear to be a need to wait for additional options. If the tree's basal flare and nebari are acceptable as you have presented to us, then my process for addressing the tree is pretty simple...
First thing before all... I would carve the deadwood on the tree... that dead face has LOTs of potential. Then I would look through the foliage on both trees for redundent and bar branches... On the ones where you have no doubt... stub them, or jin them out to the ends. If there are a bunch that need to come out, try and balance it out. Making sure to at least taper the ends so it doesn't look silly until the wood can dry and I can seriously work it over. The nice part of it is that by leaving nice jins in the canopy, you have opportunities to see through the tree (because it's entirely too dense)... and still see something interesting. Then I wire it out.
Now... this sounds like a fast process, and sometimes it is... but sometimes it's not. I always advise moving slow to examine all your options... UNLESS you are paralyzed. Then I advocate a shot of liquid courage, a pruning saw, and moonlight. (I'm not even kidding).... When I've had something beautiful that I couldn't get myself to prune, I would force myself to stop second guessing (which seems to always happen in the middle of the night on a full moon)... and I would just go cut. It may seem like a "blind" way to go at it... but in truth it isn't... you spend so many hours contemplating it you know what it looks like even in the dark. So you just decide to shut off the voice in your head and trust what you know... and you just cut. Then once "decisions" have been made... the rest is easy.
I think you acquired a marvelous tree... can't wait to see what you do with it.
Kindest regards,
Victrinia